Subject: Business Math

**LOGARITHMS**

**Introduction:**

The logarithm of a number is the index of a power, to which an already given positive base must be raised to equal the number. Introduced in the early 17^{th} century by ** John Napier**, the logarithm of a number is the exponent factor, to which the base, another fixed number, must be raised to produce that number.

For example;

The base 100 logarithm of 10000 is 2, as 100 to the power 2 is 10000,

[10000 = 100 x 100 = 100²]

i.e. 10000 = 100² or 2 = log_{100}10000

Likewise,

The base 2 logarithm of 64 is 6, as 2 to the power 6 is 64,

[64 = 2 x 2 x 2 x 2 x 2 x 2 = 2^{6}

i.e. 64 = 2^{6}or 6 = _{log64}64

Also,

64 = 4³ or 3 = log_{4}64

64 = 64¹ or 1 = log_{64}64

Therefore, the logarithms of the very same number vary from each other, in case of different bases.

**Laws of Logarithms:**

The following rules are applicable for any positive numbers x, y and e:

- log
_{e}e= 1 which follows the fact that*e¹ = e*. - log
_{e}1= 0 which follows the fact that*e*.^{0}= 1

For the following rules, assume x>0 and y>0.

- log
_{e}x^{y}= y log_{e}x - log
_{e}(xy) = log_{e}x + log_{e}y - log
_{e}(x/y) = log_{e}x - log_{e}y

**Examples:**

Find the value of:

- log
_{8}512 - log
_{√2}16

**Solution:**

- Let, x = log
_{8}512

or, 8^{x}= 512

or, 8^{x}= 8³

so, x = 3*Alternatively,*

log_{8}512 = log_{8}8³

= 3 log_{8}8.................. (log_{e}x^{y}= y log_{e}x)

= 8 x 1..................... (log_{e}e = 1)

= 8 - Let, x = log
_{√2}16or, √2

^{x}= 2^{4}or, √2

^{x}= [(√2)²]or, √2

^{x}= (√2)²so, x = 8

**Common Logarithms:**

Common logarithms are the logarithms with base 10. Almost all the numerical computations use the base 10, which is why, in the absence of the base, it is understood that the base is 10.

Such as;

10¹ = 10 or log_{10}10 = 1

10² = 100 or log_{10}10 = 2

10³ = 1000 or log_{10}10 = 3

10^{4} = 10000 or log_{10}10 = 4 and so on.

From the above examples, it can be observed that the logarithm of a number between 10 & 100 will be between 1 & 2. Therefore, it is equal to 1+a, where *a* is a positive proper fraction. Similarly, logarithms of numbers 100 & 1000 and 1000 & 10000 will be between 2 & 3 (equal to 2+b where *b* is a positive proper fraction) and 3 & 4 (equal to 3+c where *c* is a positive proper fraction) respectively and so on.

Again,

10^{-}¹ = 0.1 or log_{10}0.1 = -10.1 = -1

10^{-}² = 0.01 or log_{10}0.01 = -2 an so on.

Above examples show that the logarithm of a number between 0.1 & 0.01 will lie between -1 & -2. Hence, it is equal to -2+a where *a* is a positive proper fraction, which provides us with following conclusions:

-The logarithm of a number consists of 2 parts-

- An integer (positive, negative or zero)
- A positive proper fraction

Here, the 1^{st} part is called ** Characteristic**, whereas the 2

**Rules for finding Characteristic:**

To find characteristic, there are 2 ways. They are:

__For numbers greater than or equal to one:__“

*The characteristic of numbers greater than or equal to one is one less than the number of digits contained in its integral part.*” This is the rule that applies in the case of numbers that are greater than or equal to one.Since,

10° = 1 or, log

_{10}1 = 010¹ = 10 or, log

_{10}10 = 1The above observation shows that that the value of the logarithm of numbers between 1 & 10 is 0+a, where

*a*is the positive proper fraction. Therefore,*0*is the characteristic of numbers lying between 1 and 10.Likewise,

10² = 100 or, log

_{10}100 = 2Hence, for the numbers between 10 & 100, the logarithm value is 1+a, where

*a*is the positive proper fraction. Therefore, the characteristic of numbers lying between 1 and 10 is*1*.Similarly, characteristic of numbers lying between 100 & 1000 is

*2*and so on goes the process.

__For numbers less than one:__

“*The characteristic of numbers less than unity is negative and is one more than the number of ciphers immediately after the decimal point before any significant (i.e. non-zero) figure.*”Since,

10° = 1 or, log

_{10}1 = 010

^{-}¹ = 0.1 or, log_{10}0.1 = -1Therefore, the value of the logarithm of numbers between 0.1 & 1 is -1+a, where

*a*is the positive proper fraction. Thus, it gives the characteristic*-1*between the numbers 0.1 & 1.Likewise,

10

^{-}² = 0.01 or, log_{10}0.01 = -2So, the value of the logarithm of numbers between 0.1 & 0.01 is -2+a, where

*a*is the positive proper fraction, thus giving the characteristic*-2*and so on.Here, it is notable that in the case of a number without a cipher, immediately there is

*-1*after decimal characteristic of its logarithm. In case a number has one cipher, immediately there is*-2*after decimal characteristic of its logarithm and so goes on.

(Tamang, Pant, & G.C, 2016)

Bibliography

Tamang, G., Pant, N., & G.C, P. B. (2016). *Business Mathematics.* Putalisadak: Asmita Publication.

Logarithms

- Introduction
- Laws of Logarithms
- Common Logarithms
- Rules for finding Characteristic

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